On Wednesday, just as President Obama was giving his big speech on inequality and the economy, the Center for American Progress, a liberal think tank, released three new papers on this very topic.

Perhaps the most interesting paper here is economist Jared Bernstein's exploration (pdf) of whether rising income inequality in the United States is bad for economic growth. His conclusion: There are compelling reasons to believe that inequality can harm growth, but it's surprisingly difficult to prove this is happening.

That doesn't mean that rising inequality is benign or that there isn't a link — it's just hard to establish empirically, perhaps because of how many other factors are at play. "When you’re trying to figure out the relationship between two extremely complex variables, it’s always going to be tricky," says Bernstein, the former chief economist to Vice President Biden.

Economists used to think that income inequality was a necessary condition for growth, at least in emerging economies — the famous Kuznets curve suggests that inequality should rise sharply at first, and then the benefits of productivity become more widely shared over time. But this doesn't appear to describe the United States since the 1970s. Income inequality has soared, but the benefits of productivity growth haven't flowed as widely.

So in recent years, some economists — and even groups like the International Monetary Fund — have started wondering if high levels of inequality might even be detrimental to growth. The theories are often elegant. But, as Bernstein details in his paper, empirical support remains difficult to tease out, particularly in the case of the United States. Here's a rundown:

Theory #1: High inequality leads to under-investments in education. This was an idea put forward by economist Joseph Stiglitz. Inequality of income leads to inequality of education. Lower-income kids end up in lower-quality schools and are less able to go on to college. The result: The workforce as a whole ends up less productive overall than it would be in a more equitable economy.

But how do you prove this? It's true that wealthier families tend to spend more on educational materials — books, tutors, art and music lessons. And the gap in college completion rates between students in the top and bottom income brackets in the United States has grown over time (see chart). Growing inequality of income does seem to be leading to more inequality of education.

The trouble comes in establishing a connection between this and economic productivity. As Bernstein notes, citing an index from the Federal Reserve Bank of San Francisco, the quality of the U.S. workforce actually appears to have improved since 1979. This could be for unrelated reasons — the workforce is getting older and more experienced and overall educational opportunities are growing. So we can't disprove theory #1. But we can't yet prove it, either.

Theory #2: High inequality leads to policies that hurt growth. This is mainly a theory about politics. As income inequality grows, more and more resources are concentrated in the hands of the wealthiest. So, the idea goes, the wealthiest are better able to steer policies in directions that protect inequality at the expense of growth. (Say, high-income tax cuts instead of investments in infrastructure and R&D.) This obviously needs to be fleshed out a bit more, however — we'll return to it in a bit.

Theory #3: High inequality leads to lower levels of consumer spending. Consumer spending makes up about 70 percent of the U.S. economy. And lower-income people have a greater propensity to spend their income than wealthier people do. So, if more and more of the nation's income is concentrated at the very top, that could depress overall demand and weaken economic growth. Simple, right?

But at first glance, this doesn't hold up well, either. As the chart on the right shows, real per capita spending in the United States has continued to rise even as income inequality has gone up since 1979.

And, Bernstein notes, it's hard to find a relationship even after running simple regressions or accounting for time lags.

On the other hand, there may be more to this story. Since 2000, inequality has grown, middle-class incomes have stagnated, and poverty has stayed stubbornly high. Yet consumption kept growing — in part because there was a surge of borrowing and a housing bubble. So that brings us to a fourth theory...

Theory #4: Inequality can lead to credit bubbles and financial crises. It's also possible to combine theories #2 and #3 above and tell a story about how inequality can lead to destructive financial crises such as the one we had in 2008.

It goes like this: Incomes at the top grow. Incomes at the bottom stagnate. That creates a lot of demand for cheap credit — people at the lower end borrow more to keep afloat, while those at the top end have plenty to lend. At the same time, income inequality begins to warp politics. One example: The University of Chicago's Marianne Bertrand and Adair Morse have found that members of Congress in districts with higher levels of income inequality were more likely to vote for a 1992 bill that greatly expanded credit for housing. That was true even after controlling for ideology.

Put it all together — a surging demand for credit, a push for looser financial rules — and it's relatively easy to imagine how a nation could end up with asset bubbles and financial crises. This was a theory floated by the International Monetary Fund in 2011, which suggested that when income inequality runs rampant,  people on the lower end tend to borrow more. That excess debt, in turn, increases the odds of a major financial crisis.

Bernstein notes that there's a lot of circumstantial evidence that supports this story — including a recent paper by Barry Cynamon and Steven Fazzari establishing many of the details. "But the empirical problem, and it's a big one," Bernstein tells me, "is that you can also see many of these trends evolving in periods where there isn't as much inequality. So I'm not sure we have this one completely nailed down."

Is growth the only reason to care about inequality?

In his paper, Bernstein ultimately concludes that there still "is not enough concrete proof to lead objective observers to unequivocally conclude that inequality has held back growth," although he also notes that much of the research is "relatively new"  —  and there's a lot more work that could be done.

That said, Bernstein also points out that this isn't the only reason to care about inequality. Even if the rising gap between rich and poor didn't affect GDP in the slightest, it still might affect other things, like social mobility. For instance, he points to the research above on inequalities in education mentioned above. "Even if it doesn't affect labor quality or growth at all, you could still be extremely despairing of the kids who are facing huge barriers to meeting their academic potential," he says. "There are a lot of moving parts here."

Further reading:

--How economists have misunderstood inequality: An interview with Jamie Galbraith.

-- Here's a 2011 paper from the International Monetary Fund arguing that high levels of inequality can hurt growth. This paper is mainly a story about developing countries, however.

-- ‘Trickle-down consumption’: How rising inequality can leave everyone worse off